.999 = 1?

Factual Questions
2071

The “what I have been saying all along” : “0.999… are the ingredients to get you to the number 1.”

0.999… is not the “number”.

It is not just me, my friend. Sometimes I don’t even know what I am arguing against since other people’s idea of infinity is only that which was explained but they have no real understanding of it.

Infinity: Endless
Infinitely large: Arbitrarily Large

Then we have people saying “reaching infinity” which is clearly an oxymoron.

[QUOTE]
What is also apparent is that you do not comprehend the concept of continuity as it applies to the Reals. Of the twelve axioms that are used to define the Real number set, you have confusion over the one that distinguishes R from Q. The following confusions are exhibited:
[list][li]A failure to adequately distinguish a number from the representation of a number[/li][/QUOTE]

I know what a magnitude, value, quantity is. You require numbers to represent these items.

[QUOTE]
[li]The erroneous concept that the properties of a number depend on which base it is presented in[/li][/QUOTE]

Sometimes you can’t represent certain values in a number system adequately.
For instance:
1/3
You attempt to write 1/3 as a decimal number: 0.333… Except this is a *concept *to represent the number 1/3 and not another number.
1/3 = lim (n→inf) Σ 3/10ⁿ
However, I don’t consider : * lim (n→inf) Σ 3/10ⁿ * a number. It results in a number, but as written, it is a concept.

Also, you say the […] mean “implied limit”

Therefore: 0.333… = 0.333[…] = 0.333[limit] = limit[0.333] = 0.333
So 0.333… means: Limit 0.333 as written.

Do the ellipses mean “limit as the number of 3’s approach infinity”
Do the ellipses mean “limit as the number of 3’s equal infinity”

I assume you really mean 0.333… = lim (n→inf) Σ 3/10ⁿ
but again, I revert back to my statement that neither are numbers, but rather a process to define the number 1/3. (Construction of real numbers)

[QUOTE]
[li]the insistence that infinitesimal numbers, which do not belong to R,be treated as a non zero difference between two real numbers[/li][/QUOTE]

As (n) gets arbitrarily large, the terms themselves become infinitesimals.
Each term starts approaching 0 in Σ 3/10ⁿ

I only bring up the infinitesimal to those who are using 0.999… as a number, that is, a string of 9’s, and claim equality to 1. If you assume […] means “limit of the sequence” then we are GETTING SOMEWHERE !

[QUOTE]
[li]referencing a source that claims sqrt(2) is not a number – patently a false statement. The Pythagoreans figured this one out.[/li][/QUOTE]

Clearly a value, measure, magnitude, quantity. However, for me, numbers are a collection of digits to represent said quantity. Otherwise, you have a concept, or a function, for which you cannot calculate the number.

We are at the point where I spoke above: Considering 0.999… to be a number and 1 to be a number, and these 2 numbers are equal.
Like cos(0) [zero radians, Is that clear enough ;)]
after calculating or analyzing cos(0), we say cos(0) = 1, however is “cos(0)” a number ?? To me, it equals the number 1, but it not a number as written. It represents 1, as you like to say, but is not a number per se.

2072

I follow what you want me to think:

0.999… = lim (1 - 1/10[sup]n[/sup]) = 1

Therefore: from your quote

LHS: lim 0.999… = lim 1 = 1
RHS: 0.999… = 1

1 = 1

however, 0.999… is not a number. It is a process where the number 1 is constructed by using sequences and limits.

2073

For you, maybe, but that’s not what “number” means for everyone else here (and most everywhere else).

2074

Which brings us back to one of the other fallacies of this subject. (We really should go back over the thread and enumerate them all - this one has come up multiple times.)

The idea of “process” as a disqualification of the thing being a number carries with it the implicit notion of time. If we regard it as a definition, it can be evaluated in zero time, and clearly represents a number proper. It is only when we include some fuzzy idea of a process taking time, and thus an infinite process taking infinite time we get into trouble. There is no notion that mathematical definitions take place in the physical universe and take time. If it provides a clear definition of a number that is it - you get the number.

2075

“Evaluated” nonetheless, therefore whatever the previous representation is/was, is not a number, but a function.

2076

I assume you regard the ratio of a circle’s radius to its diameter as not being a number either.

The definition precisely and unambiguously defines a single number. What more do you want?
You might like to define such simple things as 1/2 or 2 and 3, and see how far not evaluating a function gets you.

2077

Depending on the number system, it is irrepresentable as a number.

I explained this a few times already. (value, quantity, measure, magnitude…)

0.999… was explained as a process of an infinite sum, from which we can create a sequence of partial sums, of which we can write the n[sup]th[/sup] term of (1 - 1/10[sup]n[/sup]), that of which we can take the limit of… and *finally *end up with 1.

Whether the process takes any time or not is irrelevant. “Representing an amount” and “being a number” are different things in my mind.

You *get *the number! Right. It doesn’t mean the stuff you began with *is *the number as well.
I have to go read the link that zombywoof posted.

2078

Look, consider the natural numbers 1, 2, 3…

<hangs indefinitely

2079

From zombywoof’s link (of course, wikipedia should be given relative confidence accordingly)

"Mathematical *operations *are certain *procedures *that take one or more numbers as *input *and produce a number as output.

Example: 4 + 5
Is this a *single number *as written ?? Is it the number 9 ? Or is it equal to the number 9 *after *the sum is performed ??

↑ ↑ ↑ This ↑ ↑ ↑
The “real number” as per this definition is the one that is the LIMIT… and not the “*string of digits *that is used which has a property of having a certain set of partial sums which can be used as a sequence”

My argument exactly. I would like to know what numerals these objects are:
π and √
They do not look like numerals to me. So is √2 a number, or is it a measure, which cannot be written as a number in base-10 (decimal) ??

It is for reasons like the following statement which gives me shutters about reading articles on wikipedia:
“Thus 1.0 and 0.999… are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on”
→ This is such weak sauce to convince any weak minded or average person. This comparison is total hocus pocus and not even related.
There are infinite fractions which are equal to 1.
There are infinite sequences which will converge to 1.
1.0 and 1.00 are not “different representations” of the number 1, since the number “0” is not analogous to other non-zero numbers.
1 = …0001.000…
Whether all the 0’s are written or not, doesn’t change anything. Infinite zeros are indicitive of a terminating decimal.

They also say that “0.999… (is a) decimal numeral”

So now 0.999… is a numeral ??

These are examples of the issues I have.
There are no clear “rules” laid stricly out, to avoid threads such as this.

As was said in an earlier post “… In this thread, we all interpret (such and such)…” or something along those lines. Well “this thread” is not the entire mathematical community.

I believe Dr. Carlson stated “We obviously can’t add up an infinite number of terms”
I am totally in agreement with that statement.

“Since this limit exists, we say that the sum of the series is 2 (in the shown example), even though we can’t really "do the sum.”

That is another beef of mine. We can’t do an infinite sum !! What is done in analysis is this: "…since we can’t do the sum, we will call the sum the limit"
But we know we can’t do an infinite sum! :dubious: So don’t say something that can’t be done, can be done ! It’s a contradiction !! :slight_smile:

2080

Ok, so I just researched the difference between “number” and “numeral”

Turns out there are conflicting (and opposite) definitions of their meanings.

Add that to the pile of inconsistencies.

2081

If it took 9/10 of a second to calculate the first digit and each succeeding digit takes 1/10 of the time the last one took, how long would it take?

2082

But wait - if “lim (n→inf) Σ 3/10ⁿ” is a “process” or a “concept” instead of a number, then by the same reasoning, “1/3” is a concept or process: it’s the concept of taking one and dividing it by three.

So now are you going to be consistent and also argue that 1/3 is not a number?

2083

See if this is progress:

The only way the symbols 0.999… = 1 is if you interpret the ellipses as the word “limit” and not just “an endless string of 9’s”

(You then have to remove an infinite series definition involving infinity and only use the limit of the series if you want to involve infinity. )

I believe it is using these shortcuts which have confused a lot of people, and then in the end, people think 0.999… is a “number”. (I put “number” in quotations because people use number and numeral interchangeably and a lot of the time incorrectly after my research today)

0.999… is the process by which the number **one **is constructed.
We don’t have a number “0.999…” which equals another number “one”. There is only one number here, and that number is one. It is written with the numeral 1, or can be written infinite other ways, using roman numerals, or other concepts such as cos(0), or limit (1 - 1/10[sup]n[/sup]), or … etc …

The reals, R, are the numbers which are limits of cauchy sequences.
The limit of the sequence of partial sums {9/10, 9/100, … } = 1
because the n’th term is precisely f(n) = 1 - 1/10[sup]n[/sup], and its limit is clearly 1.

Therefore, “0.999…” as written is not the number one written with arabic numerals: 1. *0.999… *is a group of other symbols and digits, which implies a process eventually ending with a limit.

Thus, believing that the “number” *0.999… *is equal to the “number” 1 is wrong. Numbers are unique.

The *process *0.999… is equal to the number “one” which can be represented as “1”, “0.999…”, cos(0), “i”, limit (1 - 1/10[sup]n[/sup]), etc…

Once you fully understand what are 0.999… and 1, what numbers and numerals are, and the construct of the actual number (not digit or numeral), can you get an appreciation of the “debate”.

And 1 more thing, using the concept of what a number is, the number 0 can certainly be represented by 0.000…1. This does not assume that “0.000…1” is a number, but a process by which we arrive at the number zero, represented by the numeral 0, and perhaps the symbols: 0.000…1, if the same understanding is used as 0.999…, (where […] mean use the limit of the sequence )
Sequence: {1/10, 1/100, 1/1000, … } or {0.1, 0.01, 0.001, … }
But remember, zero is the number, and not the sequence or an attempt to write the sequence as a string of digits.

2084

No. 1/3 is the number, as it is say, a length. The idea that 3 of them equal 1 requires no process to get “1/3”.
You can read up on ratio’s etc in Archimdes books I believe.
Also, you already used a process to get to the fraction 1/3, which was a limit of a sequence.

Lastly, it seems I had “number” and “numeral” reversed as definitions.
I admit this mistake.

PI exists as a “number” but we can’t represent it exactly with a numeral ie, a group of “digits.”
That is why another symbol was chosen, yes?

2085

Are you saying 1/3 is a “number” (in your sense) because it can be represented geometrically? You are lacking precision when you say:

What precisely do you mean 3 of them equal 1? Do you really mean when you add three of them you get one? Do you mean if you take a physical stick and break into three equal lengths this number is 1/3? You need to clarify what you mean by “no process”. Is -1 a “number”?

2086

Getting off topic. I explained my views in a previous recent comment on the topic.
About 10% of the content of my comments get addressed, then I get new questions asked before the rest are addressed, so I don’t have to make time to answer everyone else’s new questions before time is made by you to address the topics I put on the table already.

You didn’t refute what I said in the quote that you posted of mine.

2087

No, this is very much on-topic (but it seems that you’re trying to run away from your previous statements).

You claimed that 0.9999… with an endless string of nines, and that Limit 0.99999… are not numbers, they are “concepts” or “processes.”

You also made comments that the square root of two is not a number, but a process or concept.

However you’ve been referring to 1/3 as a number, and I pointed out that this is inconsistent with your other statements.

At this point, you should either admit that you should not call 1/3 a number and be consistent, or drop your previous statements about the others not being numbers.

2088

I suppose it would help if you read some recent comments:

As does sqrt(2) exist as a number by definition of number and numeral, but is not “representable” as a numeral only (only digits).

Actually I owned up to my mistake, but if it makes you feel better to think I am “running away” so you can point your finger mockingly, that is ok with me. I know what the truth is in these situations and your misunderstanding doesn’t change that truth.

I have nothing more to exchange with you because I don’t like your attitude.

Ciao :cool:

2089

Your research is a) flawed and b) irrelevant to mathematics since the inconsistencies, are inconsistencies of semantics.

In mathematics there are numbers. Numbers have definitions such as has already been described in this thread. There is for instance the number “eleven”. And the number “one third”.

Then there are numeral systems. They are used to represent numbers in writing. For positional systems the digits are often called numerals. In the decimal system the number eleven is written 11, with two instances of the numeral 1. In the octal it’s 13, with the numeral 1 and the numeral 3. And in hexadecimal it’s B, with a singel instance of the (hexadecimal) numeral B. But they’re all the same number, the number eleven.

Sometimes a numerical representation with more than one digit is also called a numeral. This is a quirk of language, not one of mathematics, and utterly irrelevant to this thread.

The number “one third” can be represented as a fraction 1/3 (or any other fraction that can be simplified to 1/3), or as a decimal expansion with “infinite decimals” 0.333… (repeating here that the definition of this notation is the limit of a series) These are the same number, the number “one third”, to pretty much any mathematician in the world. This simple mathematical notation in combination with other rigorously defined tools of mathematics then lets us examine the “scary” decimal representation 0.999… and shows us in a number of ways, many of them shown in this thread, that by any legitimate mathematical definition it equals the number one.

None of your counterarguments have held water mathematically, schooner26, and this latest adventure into the unsurprisingly non-rigid English language is irrelevant, even if you’d managed to convey it in an understandable way.

What you’re doing is pseudo"mathematics" (I added the " due to your trip into the dictionary). You started with a conclusion (one that was even based on a misunderstanding of what 0.999… means in maths) and then you ran around hunting for counterarguments, grasping at the thinnest of straws, some of which you’ve fabricated out of thin air.

2090

No, 0.99999…does not equal 1. It is not set true by definition.
It is not set true by assuming it is true. It is proved wrong.

The limit of 0.99999…is equal to 1. The limit of 0.99999…does not equal to 0.99999…

I addressed the issue [ HERE]
Furthermore
1/3 ≈ 0.3333333…